Electron cryo-microscopy of non-crystalline protein preparations (single particles) is rapidly becoming one of the most versatile and powerful techniques for the determination of the 3D structure of proteins. Current state-of-the-art electron microscopes and image processing have produced structures of protein complexes at 7A resolution. However, projections of the physical limits of this technique indicate near-atomic resolution can be achieved. The goal of this project is to develop new image processing algorithms to approach the physical limits as closely as possible. Specifically, a quantitative description of image contrast will be derived using well-studied specimens with an existing atomic model. Second, and key to high resolution, more powerful methods for the refinement of particle alignments will be developed based on maximum likelihood principles. Third, a reliable measure of resolution will be defined to serve as a hard quality measure for structures solved by single particle methods. Last, the new single particle algorithms will be put into a general form to extend their application to the alignment of unit cells in 2-dimensional crystals. The new set of programs will thus be universally applicable to ordered specimens, partially disordered specimens, and non- ordered preparations, yielding a structure at the best possible resolution in each case.